Kneser graph packing conjecture

For integers $n>k\geq 1$, let $\operatorname{KG}(n,k)$ be the graph whose vertices are the $k$-subsets of $[n]$, with two vertices adjacent exactly when the corresponding sets are disjoint. Two copies are packed into a graph when their edge sets are placed on the same vertex set without overlap. Kneser graph packing conjecture. If two copies of $\operatorname{KG}(n,k)$ can be packed into the complete graph on $\binom{n}{k}=N$ vertices, then

$$k=\Omega(n).$$

The paper proves only the weaker lower bound $k=\Omega(n/(\log n)^2)$ under the same packing hypothesis. The linear lower bound is presented as an open question related to the sunflower conjecture.